Short abstract: The problem of stabilizing the Rössler system in finite time by bounded control is considered. We employ V. I. Korobov's controllability function method, which involves a Lyapunov-type function. The controllability function is the solution of an implicit equation. A family of bounded controls which solve the problem is explicitly computed. Besides, the time that it takes the trajectory to reach the desired equilibrium is estimated.
Extended abstract: Rössler system has become one of the reference chaotic systems. Its novelty when introduced, being that exhibits a chaotic attractor generated by a simpler set of nonlinear differential equations than Lorenz system. It develops chaotic behaviour for certain values of its parameter triplet. The issue of controlling Rössler system by stabilizing one of its unstable equilibrium points has been previously dealt with in the literature. In this work, control of the Rössler system is stated by considering the synthesis problem. Given a system and one of its equilibrium points, the synthesis problem consists in constructing a bounded positional control such that for any x0 belonging to a certain neighborhood of the equilibrium point, the trajectory x(t) initiated in x0 arrives at this equilibrium point in finite time. Namely, by using V.I. Korobov’s method, also called the controllability function method, a family of bounded positional controls that solve the synthesis problem for the Rössler system is proposed. We mainly use two ingredients. The first one concerns the general theory of the controllability function The second ingredient is a family of bounded positional controls that was obtained in. Different from previous works on finite-time stabilization we propose an explicit family of bounded controls constructed by taking into account the only nonlinearity of the Rössler system, which is a quadratic function. By using the controllability function method, which is a Lyapunov-type function, the finite time to reach the desired equilibrium point is estimated. This is obtained for an arbitrary given control bound and an adequate set of initial conditions to achieve the control objective is computed. This proposal may also be developed for any controlled system for which its linear part is completely controllable and its corresponding nonlinear part is a lipschitzian function in a neighborhood of the equilibrium point. In turn, this technique may be implemented as a tool for control chaos.
Keywords: Rössler system; Korobov's controllability function; bounded control; finite time stabilization.
2010 Mathematics Subject Classification: 93C15; 93B05; 34D20. [ Full-text available (PDF) ] Top of the page.
Short abstract: In this work, the problem of generation BVI-noise by wing-shaped rotor blade of a helicopter is posed and solved. Research completed for near and far sound fields. In particular, it was discovered dependence of the distribution of density pulsations on the longitudinal blade geometry, angle of attack and blade angle to counter flow. Air flux speed increase promotes the generation of transverse pulsations on the surface blade that dominate longitudinal pulsations in level. The level of generated noise is in the range of $50\mbox{\:dB}\leq L\leq 60\mbox{\:dB}$, which is 5-6\:dB lower than the noise of the Blue Edge blades, and also rounded blade at its end.
Extended abstract: Aerodynamic noise includes a number of noise components, among which rotational noise and vortex noise (BVI-noise) make the largest contribution to the overall noise generated. Rotation noise depends on the magnitude of the velocity of the incoming blade and prevails over other noise components at significant Mach Mach numbers. Unlike rotation noise, vortex noise is evident at low helicopter flight speeds, moderate Mach numbers. In the formation of this type of noise,an important role is played by the longitudinal geometry. Therefore, recently the shape of the helicopter blade is chosen close to existing natural forms, which are as balanced as possible. One of these may be a wing-shaped blade. In this work, the problem of generating BVI noise by the wing shaped blade of a helicopter is posed and solved. The mathematical model of the problem is constructed on the previously proposed by the author and successfully tested system of aeroacoustic equations for the general case. Estimated features in this system are pulsations of sound pressure and sound potential. The calculated data of these quantities, as well as their derivatives, were used to study near and far sound fields. In particular, the dependence of the density ripple distribution is revealed from the blade geometry, the angle of attack and the blade angle to the oncoming flow. Increasing flow velocity contributes to the emergence of transverse ripples on the surface blades that dominate the longitudinal ripples by level. An interesting feature noticed in the calculations is that there are calculations for moderate Mach numbers M=0.2,0.3 situations, at certain angles of blade placement to the stream and angles of attack where rotation noise dominates eddy noise. For values Mach numbers $ M>0.4 $ rotation noise plays a major role in blade noise generation. The noise level generated is in the range $50\mbox{\:dB}\leq L\leq60\mbox{\:dB}$, which is lower by 5-6\:dB for the Blue Edge blade, as well as the rounded blade. In addition, activation of the high-frequency region in the frequency spectrum of noise was observed $f\approx{840}\mbox{\:Hz}$. The results of the calculations show that the blade of the wing-shaped is low-noise in the mode of maneuvers at small flight speeds.
Keywords: sound generation; wing-shaped helicopter blade; BVI-noise.
2010 Mathematics Subject Classification: 76Q05; 76G25. [ Full-text available (PDF) ] Top of the page.
Short abstract: It is considered the mathematical model which describes the processes of liver regeneration with homogeneous approximation. Numerical calculations revealed that the mathematical model corresponds to biological processes for different strategies of liver regeneration. Based on the calculations in the case of partial hapatectomy it is concluded that the mixed strategy of regeneration should be used for regeneration process.
Extended abstract: It is considered the generalized mathematical model which describes the processes of maintaining / restoring dynamic homeostasis (regeneration) of the liver and obviously depends on the control parameters. The model is a system of discrete controlled equations of the Lotka – Volterra type with transitions. These equations describe the controlled competitive dynamics of liver cell populations’ (hepatic lobules) various types in their various states and controlled competitive transitions between types and states. To develop this model there were accepted such assumptions: homogeneous approximation; independence of biological processes; small toxic factors. In the mathematical model the process of the liver regeneration occurs due to hyperplasia processes, replication, polyplodia and division of binuclear hepatocytes into mononuclear and controlled apoptosis. All these processes are necessary for adequate modeling of the liver regeneration. For example, single and constant toxic functions show that the above processes are not able to cope with the toxic factors that are accumulated in the body. The process of restoring the body’s functional state requires the non-trivial strategy of the liver regeneration. Numerical calculations revealed that the mathematical model corresponds to biological processes for different strategies of the liver regeneration. Based on the calculations in the case of partial hapatectomy it is concluded that the mixed strategy of regeneration should be used for the regeneration process. Henceforward it is planned to extend the mathematical model in the case of the liver regeneration, which occurs under the influence of strong toxins, that is, using the stem cells and fibrosis. It is also supposed to justify the principles and criteria for optimal regulation of the processes of maintaining / restoring liver’s dynamic homeostasis.
Keywords: mathematical model; liver regeneration; numerical experiment.
2010 Mathematics Subject Classification: 92C37, 65Y99. [ Full-text available (PDF) ] Top of the page.
2010 Mathematics Subject Classification: 01A70. [ Full-text available (PDF) ] Top of the page.
2010 Mathematics Subject Classification: 01A70. [ Full-text available (PDF) ] Top of the page.